Rheology of colloidal suspensions in confined flow: Treatment of hydrodynamic interactions in particle-based simulations inspired by dynamical density functional theory.

We investigate the microstructure and rheology of a hard-sphere suspension in a Newtonian fluid confined in a cylindrical channel and undergoing pressure-driven flow using Monte Carlo simulations. We develop a hydrodynamic framework inspired by dynamical density functional theory approaches in which the contributions due to various flow-induced hydrodynamic interactions (HI) are included in the form of thermodynamic work done by these HI-derived forces in displacing the hard spheres. Using this framework, we can self-consistently determine the effect of the local microstructure on the average flow field, and vice versa, and coevolve the inhomogeneous density distribution and the flattening velocity profile with increase in the density of suspended particles. Specifically, we explore the effect on the local microstructure due to the inclusion of forces arising from confinement-induced inertial effects, forces due to solvent-mediated interparticle interactions, and the dependence of the diffusivity on the local density. We examine the dependence of the apparent viscosity of the suspension on the volume fraction of hard spheres in the cylinder, the flow rate, and the diameter of the cylinder and investigate their effects on the local microstructure.

Microstructure of Flow-Driven Suspension of Hardspheres in Cylindrical Confinement: A Dynamical Density Functional Theory and Monte Carlo Study.

We have studied the microstructure of a flow-driven hardsphere suspension inside a cylinder using dynamical density functional theory and Monte Carlo simulations. In order to be representative of various physical conditions that may prevail in experiments, we investigate the problem using both the grand canonical (µVT) ensemble and the canonical (NVT) ensemble. In both ensembles, the hydrodynamic effect on the suspension mediated by the presence of the confining wall is implemented in a mean-field fashion by incorporating the thermodynamic work done by the inertial lift force on the particle given the average flow field. The predicted particle distribution in the µVT ensemble displays strong structural ordering at increasing flow rates due to the correspondingly higher particle concentrations inside the cylinder. In the NVT ensemble, for dilute suspensions we observe a peak in the distribution of density at a location similar to that of the Segré-Silberberg annulus, while for dense suspensions the competing effects of the inertial lift and the hardsphere interaction lead to the formation of several annuli.

Vertex stability and topological transitions in vertex models of foams and epithelia.

In computer simulations of dry foams and of epithelial tissues, vertex models are often used to describe the shape and motion of individual cells. Although these models have been widely adopted, relatively little is known about their basic theoretical properties. For example, while fourfold vertices in real foams are always unstable, it remains unclear whether a simplified vertex model description has the same behavior. Here, we study vertex stability and the dynamics of T1 topological transitions in vertex models. We show that, when all edges have the same tension, stationary fourfold vertices in these models do indeed always break up. In contrast, when tensions are allowed to depend on edge orientation, fourfold vertices can become stable, as is observed in some biological systems. More generally, our formulation of vertex stability leads to an improved treatment of T1 transitions in simulations and paves the way for studies of more biologically realistic models that couple topological transitions to the dynamics of regulatory proteins.

Energy decay in three-dimensional freely cooling granular gas.

The kinetic energy of a freely cooling granular gas decreases as a power law t(-θ) at large times t. Two theoretical conjectures exist for the exponent θ. One based on ballistic aggregation of compact spherical aggregates predicts θ=2d/(d+2) in d dimensions. The other based on Burgers equation describing anisotropic, extended clusters predicts θ=d/2 when 2≤d≤4. We do extensive simulations in three dimensions to find that while θ is as predicted by ballistic aggregation, the cluster statistics and velocity distribution differ from it. Thus, the freely cooling granular gas fits to neither the ballistic aggregation or a Burgers equation description.

Patterning the cone mosaic array in zebrafish retina requires specification of ultraviolet-sensitive cones.

Cone photoreceptors in teleost fish are organized in precise, crystalline arrays in the epithelial plane of the retina. In zebrafish, four distinct morphological/spectral cone types occupy specific, invariant positions within a regular lattice. The cone lattice is aligned orthogonal and parallel to circumference of the retinal hemisphere: it emerges as cones generated in a germinal zone at the retinal periphery are incorporated as single-cell columns into the cone lattice. Genetic disruption of the transcription factor Tbx2b eliminates most of the cone subtype maximally sensitive to ultraviolet (UV) wavelengths and also perturbs the long-range organization of the cone lattice. In the tbx2b mutant, the other three cone types (red, green, and blue cones) are specified in the correct proportion, differentiate normally, and acquire normal, planar polarized adhesive interactions mediated by Crumbs 2a and Crumbs 2b. Quantitative image analysis of cell adjacency revealed that the cones in the tbx2b mutant primarily have two nearest neighbors and align in single-cell-wide column fragments that are separated by rod photoreceptors. Some UV cones differentiate at the dorsal retinal margin in the tbx2b mutant, although they are severely dysmorphic and are eventually eliminated. Incorporating loss of UV cones during formation of cone columns at the margin into our previously published mathematical model of zebrafish cone mosaic formation (which uses bidirectional interactions between planar cell polarity proteins and anisotropic mechanical stresses in the plane of the retinal epithelium to generate regular columns of cones parallel to the margin) reproduces many features of the pattern disruptions seen in the tbx2b mutant.

Shock propagation in granular flow subjected to an external impact.

We analyze a recent experiment [Boudet, Cassagne, and Kellay, Phys. Rev. Lett. 103, 224501 (2009)] in which the shock created by the impact of a steel ball on a flowing monolayer of glass beads is studied quantitatively. We argue that radial momentum is conserved in the process and hence show that in two dimensions the shock radius increases in time t as a power law t{1/3}. This is confirmed in event driven simulations of an inelastic hard sphere system. The experimental data are compared with the theoretical prediction and are shown to compare well at intermediate times. At long times the experimental data exhibit a crossover to a different scaling behavior. We attribute this to the problem becoming effectively three dimensional due to the accumulation of particles at the shock front and propose a simple hard sphere model that incorporates this effect. Simulations of this model capture the crossover seen in the experimental data.

Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions.

Recent experiments in one- and two-dimensional microfluidic arrays of droplets containing Belousov-Zhabotinsky reactants show a rich variety of spatial patterns [M. Toiya et al., J. Phys. Chem. Lett. 1, 1241 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest-neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2π/3. We report the existence of domains with either clockwise or anticlockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.

Nonuniversal dependence of spatiotemporal regularity on randomness in coupling connections.

We investigate the spatiotemporal dynamics of a network of coupled nonlinear oscillators, modeled by sine-circle maps, with varying degrees of randomness in coupling connections. We show that the change in the basin of attraction of the spatiotemporal fixed point due to varying fraction of random links, p , is crucially related to the nature of the local dynamics. Even the qualitative dependence of the spatiotemporal regularity on p changes drastically as the angular frequency of the oscillators changes, ranging from a monotonic increase or monotonic decrease to nonmonotonic variation. Thus it is evident here that the influence of random coupling connections on spatiotemporal order is highly nonuniversal and depends very strongly on the nodal dynamics.

Spatiotemporal intermittency and scaling laws in the coupled sine circle map lattice.

We study spatiotemporal intermittency (STI) in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes with STI of both the directed percolation (DP) and non-DP class. STI with synchronized laminar behavior belongs to the DP class. The regimes of non-DP behavior show spatial intermittency (SI), where the temporal behavior of both the laminar and burst regions is regular, and the distribution of laminar lengths scales as a power law. The regular temporal behavior for the bursts seen in these regimes of spatial intermittency can be periodic or quasiperiodic, but the laminar length distributions scale with the same power law, which is distinct from the DP case. STI with traveling wave laminar states also appears in the phase diagram. Solitonlike structures appear in this regime. These are responsible for crossovers with accompanying nonuniversal exponents. The soliton lifetime distributions show power-law scaling in regimes of long average soliton lifetimes, but peak at characteristic scales with a power-law tail in regimes of short average soliton lifetimes. The signatures of each type of intermittent behavior can be found in the dynamical characterizers of the system viz. the eigenvalues of the stability matrix. We discuss the implications of our results for behavior seen in other systems which exhibit spatiotemporal intermittency.

Dynamic characterizers of spatiotemporal intermittency.

We study spatiotemporal intermittency (STI) in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes where the STI lies in the directed percolation (DP) class, as well as regimes which show pure spatial intermittency (where the temporal behavior is regular) which do not belong to the DP class. Thus both DP and non-DP behavior can be seen in the same system. The signature of DP and non-DP behavior can be seen in the dynamic characterizers, viz. the spectrum of eigenvalues of the linear stability matrix of the evolution equation, as well as in the multifractal spectrum of the eigenvalue distribution. The eigenvalue spectrum of the system in the DP regimes is continuous, whereas it shows evidence of level repulsion in the form of gaps in the spectrum in the non-DP regime. The multifractal spectrum of the eigenvalue distribution also shows the signature of DP and non-DP behavior. These results have implications for the manner in which correlations build up in extended systems.